Your ability to apply the concepts that we introduced in our previous course is enhanced when you can perform algebraic operations with matrices. At the start of this class, you will see how we can apply the Invertible Matrix Theorem to describe how a square matrix might be used to solve linear equations. This theorem is a fundamental role in linear algebra, as it synthesizes many of the concepts introduced in the first course into one succinct concept.
You will then explore theorems and algorithms that will allow you to apply linear algebra in ways that involve two or more matrices. You will examine partitioned matrices and matrix factorizations, which appear in most modern uses of linear algebra. You will also explore two applications of matrix algebra, to economics and to computer graphics.
Students taking this class are encouraged to first complete the first course in this series, linear equations.

The course is practice-oriented. It is supplemented with many problems aimed at assisting the understanding of lecture materials.
Each problem, in turn, is supplemented with a detailed solution.
The topics covered:
1. Complex algebra, complex differentiation, simple conformal mappings.
2. Taylor and Laurent expansion.
3. Residue theory. Integration of contour and real integrals with the help of residues.
4. Multivalued functions and regular branches
5. Analytic continuation and Riemann surfaces.
6. Integrals with multivalued functions.
The course includes two tracks.
The free track allows the learner to access all the materials from the course.
The "verified certificate" track allows the learner to
1. access additional non-trivial problems from the course.
2. access the detailed solutions to all the problems inside the course at the end of each week.
3. get an official certificate from the university on completion of the course.

This course by Imperial College London is designed to help you develop the skills you need to succeed in your A-level maths exams.
You will investigate key topic areas to gain a deeper understanding of the skills and techniques that you can apply throughout your A-level study. These skills include:
Fluency – selecting and applying correct methods to answer with speed and efficiency
Confidence – critically assessing mathematical methods and investigating ways to apply them
Problem solving – analysing the ‘unfamiliar’ and identifying which skills and techniques you require to answer questions
Constructing mathematical argument – using mathematical tools such as diagrams, graphs, logical deduction, mathematical symbols, mathematical language, construct mathematical argument and present precisely to others
Deep reasoning – analysing and critiquing mathematical techniques, arguments, formulae and proofs to comprehend how they can be applied
Over seven modules, covering general motion in a straight line and two dimensions, projectile motion, a model for friction, moments, equilibrium of rigid bodies, vectors, differentiation methods, integration methods and differential equations, your initial skillset will be extended to give a clear understanding of how background knowledge underpins the A -level course.
You’ll also be encouraged to consider how what you know fits into the wider mathematical world.

Introduction to linear optimization, duality and the simplex algorithm.

The goal of this mathematics course is to provide high school students and college freshmen an introduction to basic mathematics and especially show how mathematics is applied to solve fundamental engineering problems. The aim of the course is to show the students why mathematics is important in an engineering career by demonstrating how simple engineering problems can be mathematically described and methodically analyzed to find a solution.
A number of applied examples from various engineering disciplines will be introduced, analyzed and solved.

In this course, we go beyond the calculus textbook, working with practitioners in social, life and physical sciences to understand how calculus and mathematical models play a role in their work.
Through a series of case studies, you’ll learn:
How standardized test makers use functions to analyze the difficulty of test questions;
How economists model interaction of price and demand using rates of change, in a historical case of subway ridership;
How an x-ray is different from a CT-scan, and what this has to do with integrals;
How biologists use differential equation models to predict when populations will experience dramatic changes, such as extinction or outbreaks;
How the Lotka-Volterra predator-prey model was created to answer a biological puzzle;
How statisticians use functions to model data, like income distributions, and how integrals measure chance;
How Einstein’s Energy Equation, E=mc2 is an approximation to a more complicated equation.
With real practitioners as your guide, you’ll explore these situations in a hands-on way: looking at data and graphs, writing equations, doing calculus computations, and making educated guesses and predictions.
This course provides a unique supplement to a course in single-variable calculus. Key topics include application of derivatives, integrals and differential equations, mathematical models and parameters.
This course is for anyone who has completed or is currently taking a single-variable calculus course (differential and integral), at the high school (AP or IB) or college/university level. You will need to be familiar with the basics of derivatives, integrals, and differential equations, as well as functions involving polynomials, exponentials, and logarithms.
This is a course to learn applications of calculus to other fields, and NOT a course to learn the basics of calculus. Whether you’re a student who has just finished an introductory Calculus course or a teacher looking for more authentic examples for your classroom, there is something for you to learn here, and we hope you’ll join us!

Matrix Algebra underlies many of the current tools for experimental design and the analysis of high-dimensional data. In this introductory online course in data analysis, we will use matrix algebra to represent the linear models that commonly used to model differences between experimental units. We perform statistical inference on these differences. Throughout the course we will use the R programming language to perform matrix operations.
Given the diversity in educational background of our students we have divided the series into seven parts. You can take the entire series or individual courses that interest you. If you are a statistician you should consider skipping the first two or three courses, similarly, if you are biologists you should consider skipping some of the introductory biology lectures. Note that the statistics and programming aspects of the class ramp up in difficulty relatively quickly across the first three courses. You will need to know some basic stats for this course. By the third course will be teaching advanced statistical concepts such as hierarchical models and by the fourth advanced software engineering skills, such as parallel computing and reproducible research concepts.
These courses make up two Professional Certificates and are self-paced:
Data Analysis for Life Sciences:
PH525.1x: Statistics and R for the Life Sciences
PH525.2x: Introduction to Linear Models and Matrix Algebra
PH525.3x: Statistical Inference and Modeling for High-throughput Experiments
PH525.4x: High-Dimensional Data Analysis
Genomics Data Analysis:
PH525.5x: Introduction to Bioconductor
PH525.6x: Case Studies in Functional Genomics
PH525.7x: Advanced Bioconductor
This class was supported in part by NIH grant R25GM114818.

Introduction to the mathematical concept of networks, and to two important optimization problems on networks: the transshipment problem and the shortest path problem. Short introduction to the modeling power of discrete optimization, with reference to classical problems. Introduction to the branch and bound algorithm, and the concept of cuts.

Differential equations are the mathematical language we use to describe the world around us. Most phenomena can be modeled not by single differential equations, but by systems of interacting differential equations. These systems may consist of many equations. In this course, we will learn how to use linear algebra to solve systems of more than 2 differential equations. We will also learn to use MATLAB to assist us.
We will use systems of equations and matrices to explore:
The original page ranking systems used by Google,
Balancing chemical reaction equations,
Tuned mass dampers and other coupled oscillators,
Threeor more species competing for resources in an ecosystem,
The trajectory of a rider on a zipline.
The five modules in this seriesare being offered as an XSeries on edX. Please visit the
Differential EquationsXSeries Program Page
to learn more and to enroll in the modules.
*Zipline photo by teanitiki on Flickr (CC BY-SA 2.0)

More than 2000 years ago, long before rockets were launched into orbit or explorers sailed around the globe, a Greek mathematician measured the size of the Earth using nothing more than a few facts about lines, angles, and circles. This course will start at the very beginnings of geometry, answering questions like "How big is an angle?" and "What are parallel lines?" and proceed up through advanced theorems and proofs about 2D and 3D shapes. Along the way, you'll learn a few different ways to find the area of a triangle, you'll discover a shortcut for counting the number of stones in the Great Pyramid of Giza, and you'll even come up with your own estimate for the size of the Earth.
In this course, you'll be able to choose your own path within each lesson, and you can jump between lessons to quickly review earlier material. GeometryX covers a standard curriculum in high school geometry, and CCSS (common core) alignment is indicated where applicable.
Learn more about our High School and AP* Exam Preparation Courses
This course was funded in part by the Wertheimer Fund.